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Theorem lcvnbtwn3 29515
Description: The covers relation implies no in-betweenness. (cvnbtwn3 23748 analog.) (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lcvnbtwn.s  |-  S  =  ( LSubSp `  W )
lcvnbtwn.c  |-  C  =  (  <oLL  `  W )
lcvnbtwn.w  |-  ( ph  ->  W  e.  X )
lcvnbtwn.r  |-  ( ph  ->  R  e.  S )
lcvnbtwn.t  |-  ( ph  ->  T  e.  S )
lcvnbtwn.u  |-  ( ph  ->  U  e.  S )
lcvnbtwn.d  |-  ( ph  ->  R C T )
lcvnbtwn3.p  |-  ( ph  ->  R  C_  U )
lcvnbtwn3.q  |-  ( ph  ->  U  C.  T )
Assertion
Ref Expression
lcvnbtwn3  |-  ( ph  ->  U  =  R )

Proof of Theorem lcvnbtwn3
StepHypRef Expression
1 lcvnbtwn3.p . 2  |-  ( ph  ->  R  C_  U )
2 lcvnbtwn3.q . 2  |-  ( ph  ->  U  C.  T )
3 lcvnbtwn.s . . . 4  |-  S  =  ( LSubSp `  W )
4 lcvnbtwn.c . . . 4  |-  C  =  (  <oLL  `  W )
5 lcvnbtwn.w . . . 4  |-  ( ph  ->  W  e.  X )
6 lcvnbtwn.r . . . 4  |-  ( ph  ->  R  e.  S )
7 lcvnbtwn.t . . . 4  |-  ( ph  ->  T  e.  S )
8 lcvnbtwn.u . . . 4  |-  ( ph  ->  U  e.  S )
9 lcvnbtwn.d . . . 4  |-  ( ph  ->  R C T )
103, 4, 5, 6, 7, 8, 9lcvnbtwn 29512 . . 3  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )
11 iman 414 . . . 4  |-  ( ( ( R  C_  U  /\  U  C.  T )  ->  R  =  U )  <->  -.  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
12 eqcom 2410 . . . . 5  |-  ( U  =  R  <->  R  =  U )
1312imbi2i 304 . . . 4  |-  ( ( ( R  C_  U  /\  U  C.  T )  ->  U  =  R )  <->  ( ( R 
C_  U  /\  U  C.  T )  ->  R  =  U ) )
14 dfpss2 3396 . . . . . . 7  |-  ( R 
C.  U  <->  ( R  C_  U  /\  -.  R  =  U ) )
1514anbi1i 677 . . . . . 6  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( ( R  C_  U  /\  -.  R  =  U )  /\  U  C.  T ) )
16 an32 774 . . . . . 6  |-  ( ( ( R  C_  U  /\  -.  R  =  U )  /\  U  C.  T )  <->  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
1715, 16bitri 241 . . . . 5  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
1817notbii 288 . . . 4  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <->  -.  ( ( R  C_  U  /\  U  C.  T
)  /\  -.  R  =  U ) )
1911, 13, 183bitr4ri 270 . . 3  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <-> 
( ( R  C_  U  /\  U  C.  T
)  ->  U  =  R ) )
2010, 19sylib 189 . 2  |-  ( ph  ->  ( ( R  C_  U  /\  U  C.  T
)  ->  U  =  R ) )
211, 2, 20mp2and 661 1  |-  ( ph  ->  U  =  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3284    C. wpss 3285   class class class wbr 4176   ` cfv 5417   LSubSpclss 15967    <oLL clcv 29505
This theorem is referenced by:  lsatcveq0  29519  lsatcvatlem  29536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-lcv 29506
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